A direct O(N log2 N) finite difference method for fractional diffusion equations

Fractional diffusion equations model phenomena exhibiting anomalous diffusion that can not be modeled accurately by the second-order diffusion equations. Because of the nonlocal property of fractional differential operators, the numerical methods have full coefficient matrices which require storage of O(N^2) and computational cost of O(N^3) where N is the number of grid points. In this paper we develop a fast finite difference method for fractional diffusion equations, which only requires storage of O(N) and computational cost of O(Nlog^2N) while retaining the same accuracy and approximation property as the regular finite difference method. Numerical experiments are presented to show the utility of the method.

[1]  Richard S. Varga,et al.  Matrix Iterative Analysis , 2000, The Mathematical Gazette.

[2]  Zhaoxia Yang,et al.  Finite difference approximations for the fractional advection-diffusion equation , 2009 .

[3]  D. Benson,et al.  The fractional‐order governing equation of Lévy Motion , 2000 .

[4]  A. Böttcher,et al.  Introduction to Large Truncated Toeplitz Matrices , 1998 .

[5]  Stevens,et al.  Self-similar transport in incomplete chaos. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[6]  X. Li,et al.  Existence and Uniqueness of the Weak Solution of the Space-Time Fractional Diffusion Equation and a Spectral Method Approximation , 2010 .

[7]  M. Meerschaert,et al.  Finite difference approximations for two-sided space-fractional partial differential equations , 2006 .

[8]  M. Meerschaert,et al.  Finite difference approximations for fractional advection-dispersion flow equations , 2004 .

[9]  Chuanju Xu,et al.  Finite difference/spectral approximations for the time-fractional diffusion equation , 2007, J. Comput. Phys..

[10]  J. Kirchner,et al.  Fractal stream chemistry and its implications for contaminant transport in catchments , 2000, Nature.

[11]  Mark M. Meerschaert,et al.  A second-order accurate numerical approximation for the fractional diffusion equation , 2006, J. Comput. Phys..

[12]  Robert M. Gray,et al.  Toeplitz and Circulant Matrices: A Review , 2005, Found. Trends Commun. Inf. Theory.

[13]  Peter Richmond,et al.  Waiting time distributions in financial markets , 2002 .

[14]  Vickie E. Lynch,et al.  Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model , 2001 .

[15]  M. Meerschaert,et al.  Finite difference methods for two-dimensional fractional dispersion equation , 2006 .

[16]  Mihály Kovács,et al.  Numerical solutions for fractional reaction-diffusion equations , 2008, Comput. Math. Appl..

[17]  V. Ervin,et al.  Variational formulation for the stationary fractional advection dispersion equation , 2006 .

[18]  Diego A. Murio,et al.  Implicit finite difference approximation for time fractional diffusion equations , 2008, Comput. Math. Appl..

[19]  I. Podlubny.  Fractional differential equations , 1998 .

[20]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[21]  A. Chaves,et al.  A fractional diffusion equation to describe Lévy flights , 1998 .

[22]  Enrico Scalas,et al.  Waiting-times and returns in high-frequency financial data: an empirical study , 2002, cond-mat/0203596.

[23]  Weihua Deng,et al.  Finite Element Method for the Space and Time Fractional Fokker-Planck Equation , 2008, SIAM J. Numer. Anal..

[24]  J. Klafter,et al.  The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics , 2004 .

[25]  R. Magin.  Fractional Calculus in Bioengineering , 2006 .

[26]  Bruce J. West,et al.  Lévy dynamics of enhanced diffusion: Application to turbulence. , 1987, Physical review letters.

[27]  Fawang Liu,et al.  Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation , 2009, Appl. Math. Comput..

[28]  Norbert Heuer,et al.  Numerical Approximation of a Time Dependent, Nonlinear, Space-Fractional Diffusion Equation , 2007, SIAM J. Numer. Anal..

[29]  Mingrong Cui,et al.  Compact finite difference method for the fractional diffusion equation , 2009, J. Comput. Phys..

[30]  Fawang Liu,et al.  Numerical solution of the space fractional Fokker-Planck equation , 2004 .

[31]  D. Benson,et al.  Eulerian derivation of the fractional advection-dispersion equation. , 2001, Journal of contaminant hydrology.

[32]  D. Benson,et al.  Application of a fractional advection‐dispersion equation , 2000 .

[33]  Christine Doughty.  Investigation of conceptual and numerical approaches for evaluating moisture, gas, chemical, and heat transport in fractured unsaturated rock , 1999 .

[34]  Ercília Sousa,et al.  Finite difference approximations for a fractional advection diffusion problem , 2009, J. Comput. Phys..

[35]  B. Henry,et al.  The accuracy and stability of an implicit solution method for the fractional diffusion equation , 2005 .

[36]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[37]  Yang Zhang,et al.  A finite difference method for fractional partial differential equation , 2009, Appl. Math. Comput..